Chaos Theories Visuals

Math Chaos

Chaos Theory Visualizations

This project provides a suite of Python tools for computing and graphing classic systems from chaos theory, specifically focusing on the Lorenz Attractor and the Logistic Map.

Featured Systems

3D Lorenz Attractor

A 3D representation of the Lorenz system, showcasing the famous butterfly-shaped strange attractor.

2D Axis Projections

Detailed 2D graphs of the Lorenz attractor’s trajectories across the XY, YZ, and XZ planes.

Logistic Map

Visualization of the logistic map and its bifurcation diagram, illustrating how simple non-linear equations lead to chaos.

Interactive Parameters

Customizable system parameters (sigma, beta, rho) to explore different chaotic regimes.

Theoretical Context

Lorenz System

The Lorenz system is a set of three ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions.

dx/dt = sigma * (y - x)
dy/dt = x * (rho - z) - y
dz/dt = x * y - beta * z

Logistic Map

The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behavior can arise from very simple non-linear dynamical equations.

Getting Started

Prerequisites

Ensure you have Python installed along with the following libraries:

numpy
matplotlib
scipy

Install them using pip:

pip install -r requirements.txt

Usage

Configure Parameters Open parameters.py and set your desired values for sigma, beta, rho, and time.
Run a Simulation Execute the desired script from the terminal:
Terminal window
```
python 3D_Lorenz_Attractor.py
```
or
Terminal window
```
python logisticmap.py
```

Technical Details

Language: Python 3.x
Visualization: matplotlib for 2D and 3D plotting.
Numerical Solver: scipy for integrating the differential equations.